Limit Calculator
Evaluate lim f(x) as x approaches any finite value or infinity.
Infinity for ∞. Example: (x^2 - 1)/(x - 1) as x approaches 1.Limit Calculator: Approach, Continuity & Convergence
The Limit Calculator evaluates the behavior of a function as x approaches a specific value, including positive or negative infinity. Limits underpin the formal definitions of both derivatives and integrals. Our tool computes left-hand, right-hand, and two-sided limits via high-precision numerical approximation (h = 1×10-8).
One-Sided vs Two-Sided Limits
A two-sided limit lim[x→a] f(x) exists only when the left-hand and right-hand limits are equal. If they differ, the limit does not exist (DNE). This matters for piecewise functions, absolute values, and rational functions with holes or vertical asymptotes.
Limits at Infinity
When x → ±∞, limits describe horizontal asymptotes — the long-run behavior of a function. For rational functions, compare degrees of numerator and denominator. Our tool substitutes x = ±1010 for reliable approximations.
L'Hopital's Rule
When direct substitution produces 0/0 or ∞/∞, L'Hopital's Rule applies: lim f/g = lim f′/g′. Our numerical engine resolves these indeterminate forms automatically by approaching with infinitesimally small increments.